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@@ -1,142 +1,142 @@
----
-title: Hand-wave Quantum Solver
-emoji: 🌊
-colorFrom: blue
-colorTo: purple
-sdk: streamlit
-sdk_version: "1.28.0"
-app_file: app.py
-pinned: false
----
-
-# ⚛️ Quantum Potential Solver
-
-**Author:** Ahilan Kumaresan  
-**Institution:** Simon Fraser University  
-**Field:** Mathematical & Computational Physics
-
----
-
-## 🎯 Overview
-
-An advanced numerical solver for the **Time-Independent Schrödinger Equation** (TISE) using the Finite Difference Method. This project demonstrates rigorous computational physics methodology with comprehensive verification against analytical solutions and external libraries.
-
-## 🔬 Key Features
-
-### 1. **Accurate Numerical Solver**
-- **Method:** 3-point Central Difference stencil for the Laplacian operator
-- **Grid:** Adaptive resolution (1000-2000 points)
-- **Units:** Hartree Atomic Units (ℏ=1, m=1)
-- **Boundary Conditions:** Dirichlet (infinite walls)
-
-### 2. **Multiple Potential Types**
-- Infinite Square Well
-- Harmonic Oscillator
-- Double Well Potential
-- Custom potentials via hand gestures (camera input)
-
-### 3. **Interactive Visualizations**
-- Plotly-based interactive charts
-- Wavefunction probability densities
-- Energy level diagrams
-- Real-time solver performance metrics
-
-### 4. **Rigorous Verification**
-
-#### Analytical Benchmarks
-| System | Max Error |
-|--------|-----------|
-| Infinite Square Well | < 0.003% |
-| Harmonic Oscillator | < 0.02% |
-| Half-Harmonic Oscillator | < 0.8% |
-| Triangular Potential | < 0.003% |
-
-#### External Library Comparison
-- **Cross-verified with QMSolve** (Python quantum mechanics package)
-- **Double Well Potential:** Agreement within 0.25%
-- Demonstrates accuracy for systems without analytical solutions
-
-## 📐 Mathematical Foundation
-
-The solver discretizes the TISE:
-
-$$\hat{H}\psi(x) = E\psi(x)$$
-
-$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) \right]\psi(x) = E\psi(x)$$
-
-Using finite differences:
-
-$$\frac{d^2\psi}{dx^2} \approx \frac{\psi_{i+1} - 2\psi_i + \psi_{i-1}}{\Delta x^2}$$
-
-This transforms the problem into a matrix eigenvalue equation solved via `numpy.linalg.eigh`.
-
-## 🛠️ Technical Implementation
-
-- **Language:** Python 3.12
-- **Core Libraries:** NumPy, SciPy
-- **Visualization:** Plotly, Matplotlib
-- **UI Framework:** Streamlit
-- **Computer Vision:** MediaPipe (for hand gesture input)
-
-## 📊 Verification Scripts
-
-The project includes comprehensive verification:
-- `verify_physics.py`: Analytical benchmarks
-- `verify_comparison.py`: QMSolve cross-verification
-- `Comparison_Notebook.ipynb`: Jupyter notebook with detailed analysis
-
-## 🎓 Academic Applications
-
-This project demonstrates:
-1. **Numerical Methods:** Finite difference discretization of differential operators
-2. **Linear Algebra:** Eigenvalue problems for Hermitian matrices
-3. **Quantum Mechanics:** Stationary states and energy quantization
-4. **Software Engineering:** Modular design, comprehensive testing, professional documentation
-
-## 📖 Usage
-
-### Interactive Simulator
-1. Select a potential type from the sidebar
-2. Adjust parameters using sliders
-3. View real-time solutions with interactive plots
-
-### Verification Dashboard
-- Navigate to "Benchmarks & Verification" to see accuracy metrics
-- Compare against analytical solutions and QMSolve
-
-### Theory Section
-- Detailed mathematical background
-- Implementation methodology
-- Code verification
-
-## 🔗 Repository Structure
-
-```
-psi_solve2/
-├── app.py                      # Main Streamlit application
-├── functions.py                # Physics engine (solver + potentials)
-├── verify_physics.py           # Analytical verification script
-├── verify_comparison.py        # QMSolve comparison script
-├── Comparison_Notebook.ipynb   # Jupyter analysis notebook
-└── requirements.txt            # Python dependencies
-```
-
-## 📝 Citation
-
-If you use this solver in your research, please cite:
-
-```
-Kumaresan, A. (2024). Quantum Potential Solver: A Verified Finite Difference 
-Implementation for the Time-Independent Schrödinger Equation. 
-Simon Fraser University.
-```
-
-## 📧 Contact
-
-**Ahilan Kumaresan**  
-Mathematical & Computational Physics  
-Simon Fraser University
-
----
-
-*This project showcases advanced computational physics methodology suitable for graduate-level research in quantum mechanics and numerical analysis.*
+---
+title: Hand-wave Quantum Solver
+emoji: 🌊
+colorFrom: blue
+colorTo: purple
+sdk: streamlit
+sdk_version: "1.28.0"
+app_file: app.py
+pinned: false
+---
+
+# ⚛️ Quantum Potential Solver
+
+**Author:** Ahilan Kumaresan  
+**Institution:** Simon Fraser University  
+**Field:** Mathematical & Computational Physics
+
+---
+
+## 🎯 Overview
+
+An advanced numerical solver for the **Time-Independent Schrödinger Equation** (TISE) using the Finite Difference Method. This project demonstrates rigorous computational physics methodology with comprehensive verification against analytical solutions and external libraries.
+
+## 🔬 Key Features
+
+### 1. **Accurate Numerical Solver**
+- **Method:** 3-point Central Difference stencil for the Laplacian operator
+- **Grid:** Adaptive resolution (1000-2000 points)
+- **Units:** Hartree Atomic Units (ℏ=1, m=1)
+- **Boundary Conditions:** Dirichlet (infinite walls)
+
+### 2. **Multiple Potential Types**
+- Infinite Square Well
+- Harmonic Oscillator
+- Double Well Potential
+- Custom potentials via hand gestures (camera input)
+
+### 3. **Interactive Visualizations**
+- Plotly-based interactive charts
+- Wavefunction probability densities
+- Energy level diagrams
+- Real-time solver performance metrics
+
+### 4. **Rigorous Verification**
+
+#### Analytical Benchmarks
+| System | Max Error |
+|--------|-----------|
+| Infinite Square Well | < 0.003% |
+| Harmonic Oscillator | < 0.02% |
+| Half-Harmonic Oscillator | < 0.8% |
+| Triangular Potential | < 0.003% |
+
+#### External Library Comparison
+- **Cross-verified with QMSolve** (Python quantum mechanics package)
+- **Double Well Potential:** Agreement within 0.25%
+- Demonstrates accuracy for systems without analytical solutions
+
+## 📐 Mathematical Foundation
+
+The solver discretizes the TISE:
+
+$$\hat{H}\psi(x) = E\psi(x)$$
+
+$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) \right]\psi(x) = E\psi(x)$$
+
+Using finite differences:
+
+$$\frac{d^2\psi}{dx^2} \approx \frac{\psi_{i+1} - 2\psi_i + \psi_{i-1}}{\Delta x^2}$$
+
+This transforms the problem into a matrix eigenvalue equation solved via `numpy.linalg.eigh`.
+
+## 🛠️ Technical Implementation
+
+- **Language:** Python 3.12
+- **Core Libraries:** NumPy, SciPy
+- **Visualization:** Plotly, Matplotlib
+- **UI Framework:** Streamlit
+- **Computer Vision:** MediaPipe (for hand gesture input)
+
+## 📊 Verification Scripts
+
+The project includes comprehensive verification:
+- `verify_physics.py`: Analytical benchmarks
+- `verify_comparison.py`: QMSolve cross-verification
+- `Comparison_Notebook.ipynb`: Jupyter notebook with detailed analysis
+
+## 🎓 Academic Applications
+
+This project demonstrates:
+1. **Numerical Methods:** Finite difference discretization of differential operators
+2. **Linear Algebra:** Eigenvalue problems for Hermitian matrices
+3. **Quantum Mechanics:** Stationary states and energy quantization
+4. **Software Engineering:** Modular design, comprehensive testing, professional documentation
+
+## 📖 Usage
+
+### Interactive Simulator
+1. Select a potential type from the sidebar
+2. Adjust parameters using sliders
+3. View real-time solutions with interactive plots
+
+### Verification Dashboard
+- Navigate to "Benchmarks & Verification" to see accuracy metrics
+- Compare against analytical solutions and QMSolve
+
+### Theory Section
+- Detailed mathematical background
+- Implementation methodology
+- Code verification
+
+## 🔗 Repository Structure
+
+```
+psi_solve2/
+├── app.py                      # Main Streamlit application
+├── functions.py                # Physics engine (solver + potentials)
+├── verify_physics.py           # Analytical verification script
+├── verify_comparison.py        # QMSolve comparison script
+├── Comparison_Notebook.ipynb   # Jupyter analysis notebook
+└── requirements.txt            # Python dependencies
+```
+
+## 📝 Citation
+
+If you use this solver in your research, please cite:
+
+```
+Kumaresan, A. (2024). Quantum Potential Solver: A Verified Finite Difference 
+Implementation for the Time-Independent Schrödinger Equation. 
+Simon Fraser University.
+```
+
+## 📧 Contact
+
+**Ahilan Kumaresan**  
+Mathematical & Computational Physics  
+Simon Fraser University
+
+---
+
+*This project showcases advanced computational physics methodology suitable for graduate-level research in quantum mechanics and numerical analysis.*